Sometimes you really like someone, but you can't for the life of you understand why. By all means, you should have tired of them long ago, but you keep coming back for more. Welcome, my friend, to Topology.
This book is a good one, but boy was it slow (349 pages at ~30 minutes a page, on average). I just kept coming back, and I was slowly rewarded each time I did.
Note: sil ver already reviewed Topology.
Topology is about what it means for things to be "close" in a very abstract and general sense. Rather than taking on the monstrous task of intuitively explaining topology without math, I'm just going to talk about random things from the book and (literally) illustrate concepts which were at first confusing.
Compactness = wonderful kind of mathematical "smallness"
Compact means small. It is a peculiar kind of small, but at its heart, compactness is a precise way of being small in the mathematical world. The smallness is peculiar because, as in the example of the open and closed intervals and , a set can be made “smaller” (that is, compact) by adding points to it, and it can be made “larger” (non-compact) by taking points away.
As a notion of smallness, then, compactness is a bit fraught. It’s a bit unsettling to say that a set can be “smaller” than a set that lies entirely inside it! But I think smallness is a valuable way to see compactness. A set that is compact may be large in area and complicated, but the fact that it is compact means we can interact with it in a finite way using open sets, the building blocks of topology.
Minimum description length says that an explanation is big if its shortest computational specification is long. You can have a simple explanation of a very long list of things or of a large universe, and extremely complicated explanations of things easily expressed in natural language (God's source code would be a lot longer than Maxwell's equations).
VC dimension says a class of hypotheses is hard to learn if it has lots of predictive degrees of freedom. You can have an infinite class of hypotheses which is really easy to learn because it has low VC dimension (thresholding functions at value ), and a finite class which is really hard to learn because it has high VC dimension (all C programs less than 1 million characters).
Compactness says that a topological space is big if it has a covering of open sets that can't be trimmed down to a finite subcollection which still covers the whole space. You can have an uncountable compact space ( under the standard topology, or even a Cantor space), and a countable space which isn't compact ( under the standard topology; note that all countable topological spaces have to at least be Lindelof).
Compactness is not always inherited by open subspaces
At first, I was confused why open subspaces of compact don't have to be compact (if is closed, it does have to be compact). But compactness requires all open coverings of to have a finite subcover. Meaning, you can't just give it 's finite cover intersect the subspace, because the finite subcover has to be a subcollection of 's covering.
Theorem: If is compact, show that the projection is closed.
I was confused why we needed compactness. Essentially, I didn't understand the tube lemma.
Now let's prove the theorem. Suppose is closed in . We want to show is also closed. Take . is an open set of the domain containing the slice . Since is compact, apply the tube lemma to get a tube . The projection of this tube is both open (because is open in ) and disjoint from (because the tube is contained in ). Thus, all have an open neighborhood disjoint from , so must be closed.
Let be a locally compact space. If is continuous, does it follow that is locally compact? What if is both continuous and open?
It has to be both continuous and open; the reason I got confused here was it seemed like continuity should be enough. It was plain to me how to prove it given open, but this SE post has a good counterexample for just continuous.
How come you can have discontinuous multivariate functions which are continuous in each variable? What is continuity, with a product space as your domain? To simplify matters, let’s consider two metric spaces .
One definition of continuity uses open sets – is continuous at if, for every open neighborhood of , there exists an open neighborhood of such that .
Another definition uses topological convergence. is continuous at if, for every sequence , .
These definitions are equivalent. The latter lets us think about how different winding paths you can take in a domain always must topologically converge to the same thing in the co-domain.
Continuity in the variables says that paths along the axes converge in the right way. But for continuity overall, we need all paths to converge in the right way. Directional continuity when the domain is is a special case of this: continuity from below and from above if and only if continuity for all sequences converging topologically to .
You only lift once
Once you fix the initial point, the lift corresponds to the unique path in the covering space which produces . It's just helping you find the corresponding path in the lifted up covering space!
This concept yields amazing insight into such profound topics as the deeper nature of jump rope. Under the standard subspace topology of , consider the space swept out by a rope held at fixed endpoints and tautness. All paths between the endpoints are path homotopic! You can think about movements of the rope (either clockwise or counterclockwise) as homotopies in this space.
I stopped at about section 56 because I was getting diminishing returns. By this point, I felt like I had a solid understanding of point-set topology, and look forward to more thoroughly covering algebraic topology in the future.
One-point compactifications feel like an important thing to grasp, and they're fun to play around with mentally. I skipped Stone-Cech compactification.
Completeness in metric spaces means that Cauchy sequences converge topologically; in other words, nothing can "escape" from the space. I remember having problems with this (and with thinking about non-Hausdorff spaces) back when I was learning analysis. Things feel a lot better now.
Topology can be dry, but it's exceedingly well-written and clear. I tried for quite a while to find a better topology book, but I didn't.
Finally getting around to topology was such a good decision. For exercise solutions, see both MathOverflow and this site.
Some things change how you look at math, help you notice subtleties and shades and immediately grasp certain facets of new mathematical objects. Topology is one of these things, as is abstract algebra. Learning that an object is a group, or finitely generated, or isomorphic to a more familiar structure gives me an immediate head start. Similarly, learning that spaces are homeomorphic, or compact, or second-countable is such a boost.
What was I even doing with my life before I knew about homeomorphisms?